While the celebrated graph neural networks (GNNs) yield effective representations for individual nodes of a graph, there has been relatively less success in extending to the task of graph similarity learning. Recent work on graph similarity learning has considered either global-level graph-graph interactions or low-level node-node interactions, however, ignoring the rich cross-level interactions (e.g., between each node of one graph and the other whole graph). In this article, we propose a multilevel graph matching network (MGMN) framework for computing the graph similarity between any pair of graph-structured objects in an end-to-end fashion. In particular, the proposed MGMN consists of a node-graph matching network (NGMN) for effectively learning cross-level interactions between each node of one graph and the other whole graph, and a siamese GNN to learn global-level interactions between two input graphs. Furthermore, to compensate for the lack of standard benchmark datasets, we have created and collected a set of datasets for both the graph-graph classification and graph-graph regression tasks with different sizes in order to evaluate the effectiveness and robustness of our models. Comprehensive experiments demonstrate that MGMN consistently outperforms state-of-the-art baseline models on both the graph-graph classification and graph-graph regression tasks. Compared with previous work, multilevel graph matching network (MGMN) also exhibits stronger robustness as the sizes of the two input graphs increase.